Find the coordinates of a parallelogram from vector information

Expert intro: Once one vertex and two adjacent side vectors are known, the rest of the parallelogram follows by vector addition. It is mostly bookkeeping, but the signs need attention.

Detailed walkthrough

We are given

$\overrightarrow{OA}=2\mathbf i-\mathbf j=(2,-1),$ $\overrightarrow{AB}=3\mathbf i+4\mathbf j=(3,4),$ $\overrightarrow{AD}=-2\mathbf i+5\mathbf j=(-2,5).$

Since $O$ is the origin, the coordinates of $A$ are directly

$A=(2,-1).$

Now use vector addition.

Find $B$

$\overrightarrow{OB}=\overrightarrow{OA}+\overrightarrow{AB}.$

So

$(2,-1)+(3,4)=(5,3),$

hence

$B=(5,3).$

Find $D$

$\overrightarrow{OD}=\overrightarrow{OA}+\overrightarrow{AD}.$

So

$(2,-1)+(-2,5)=(0,4),$

hence

$D=(0,4).$

Find $C$

In a parallelogram, $\overrightarrow{BC}=\overrightarrow{AD}$ or equivalently

$\overrightarrow{OC}=\overrightarrow{OB}+\overrightarrow{BC}.$

Thus

$(5,3)+(-2,5)=(3,8),$

so

$C=(3,8).$

Therefore the four vertices are

$\boxed{A(2,-1),\ B(5,3),\ C(3,8),\ D(0,4)}.$

💡 Pitfall guide

A frequent mistake is mixing up $\overrightarrow{AB}$ with the position vector of $B$. Remember: $\overrightarrow{OB}=\overrightarrow{OA}+\overrightarrow{AB}$, not just $\overrightarrow{AB}$. Also check the order of vertices; in a parallelogram, $C$ must be reached consistently from either $B$ or $D$.

🔄 Real-world variant

If the origin were not given, you would first need the position vector of one vertex or a coordinate reference point. If only $A$, $\overrightarrow{AB}$, and $\overrightarrow{AD}$ were known, then the whole shape could still be built, but the final coordinates would be relative to the chosen origin.

🔍 Related terms

position vector, parallelogram, vector addition